Optimal timing of arrival to a queue

Excerpt

Suppose a central planner (CP) wishes to schedule the arrival of \(n \ge 2\) customers to a single-server queue during the time interval [0, T]. Service times are independent and follow an exponential distribution with parameter \(\mu \). Service can go on, if not completed, beyond time T. The CP’s objective is to minimize the expected total waiting time. Observe that our assumptions so far and the stated objective assume homogeneity among customers. Assume that the CP cannot tell each individual when to arrive but rather needs to broadcast to all the same arrival strategy. This is commonly the case where the actual participants are unknown, where their number is large and/or when they vary from time to time. Yet, he can announce some distribution for their arrival time and then all will follow it independently. Thus, the question is what should be the optimal distribution. It is easy to see that a pure strategy which tells all to arrive at some (in fact, any) time is certainly not optimal. Such problems come naturally in when-to-arrive models. Possible examples are the concert hall or cafeteria models as well as modeling patients arriving to a clinic during a given day (see [6, 7] or [3]). A recent survey of the literature of when-to-arrive appears in [5]. …